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Theorem ereldm 8748
Description: Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ereldm.1 (𝜑𝑅 Er 𝑋)
ereldm.2 (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
Assertion
Ref Expression
ereldm (𝜑 → (𝐴𝑋𝐵𝑋))

Proof of Theorem ereldm
StepHypRef Expression
1 ereldm.2 . . . 4 (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
21neeq1d 2992 . . 3 (𝜑 → ([𝐴]𝑅 ≠ ∅ ↔ [𝐵]𝑅 ≠ ∅))
3 ecdmn0 8747 . . 3 (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅)
4 ecdmn0 8747 . . 3 (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅)
52, 3, 43bitr4g 314 . 2 (𝜑 → (𝐴 ∈ dom 𝑅𝐵 ∈ dom 𝑅))
6 ereldm.1 . . . 4 (𝜑𝑅 Er 𝑋)
7 erdm 8710 . . . 4 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
86, 7syl 17 . . 3 (𝜑 → dom 𝑅 = 𝑋)
98eleq2d 2811 . 2 (𝜑 → (𝐴 ∈ dom 𝑅𝐴𝑋))
108eleq2d 2811 . 2 (𝜑 → (𝐵 ∈ dom 𝑅𝐵𝑋))
115, 9, 103bitr3d 309 1 (𝜑 → (𝐴𝑋𝐵𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  wne 2932  c0 4315  dom cdm 5667   Er wer 8697  [cec 8698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-er 8700  df-ec 8702
This theorem is referenced by:  erth  8749  brecop  8801  eceqoveq  8813
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